##### Propositional Logic

Introduction to Reasoning

• tautology
• contingency

### Introduction to Reasoning

Logical reasoning is the process of drawing conclusions from premises using rules of inference. Here we are going to study reasoning with propositions. Later we are going to see reasoning with predicate logic, which allows us to reason about individual objects. However, inference rules of propositional logic are also applicable to predicate logic and reasoning with propositions is fundamental to reasoning with predicate logic.
These inference rules are results of observations of human reasoning over centuries. Though there is nothing absolute about them, they have contributed significantly in the scientific and engineering progress the mankind have made. Today they are universally accepted as the rules of logical reasoning and they should be followed in our reasoning.

Since inference rules are based on identities and implications, we are going to study them first. We start with three types of proposition which are used to define the meaning of "identity" and "implication".

### Types of Proposition

Some propositions are always true regardless of the truth value of its component propositions.
For example (P P) is always true regardless of the value of the proposition P.
A proposition that is always true called a tautology.
There are also propositions that are always false such as (P P).
Such a proposition is called a contradiction.
A proposition that is neither a tautology nor a contradiction is called a contingency.
For example (P Q) is a contingency.

These types of propositions play a crucial role in reasoning. In particular every inference rule is a tautology as we see in identities and implications.